Lie dialgebras, gauge theory, and Lagrangian multiforms

Much of my current work is rooted in the r-matrix approach to integrable systems, the framework of holomorphic-topological gauge theory, and the theory of Lagrangian multiforms. Lagrangian multiforms provide a variational framework for describing integrable hierarchies. During my doctoral studies at the University of Leeds, together with my doctoral supervisors and our collaborators, I developed two approaches for systematically constructing Lagrangian one-forms, which cover the case of finite-dimensional integrable hierarchies, thus addressing one of the central open problems in the theory of Lagrangian multiforms.

The first of these approaches, developed in the following two works, is based on the theory of Lie dialgebras and incorporates into Lagrangian one-forms the notion of the classical r-matrix and produces Lagrangian one-forms living on coadjoint orbits.

Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
Vincent Caudrelier, Marta Dell’Atti, and Anup Anand Singh
Letters in Mathematical Physics
arXiv:2307.07339 [math-ph]
February 2024

Lagrangian multiform for cyclotomic Gaudin models
Vincent Caudrelier, Anup Anand Singh, and Benoît Vicedo
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
arXiv:2405.12837 [math-ph]
November 2024

In a more recent work where the second approach was developed, my collaborators and I derived a Lagrangian multiform for the Hitchin system, establishing a connection between 3d holomorphic-topological BF theory and the Hitchin system at the classical level in the process. The Hitchin system is related to vector bundles on Riemann surfaces and unifies many interesting integrable models.

The 3d mixed BF Lagrangian 1-form: a variational formulation of Hitchin’s integrable system
Vincent Caudrelier, Derek Harland, Anup Anand Singh, and Benoît Vicedo
Communications in Mathematical Physics
arXiv:2509.05127 [math-ph]
January 2026

A copy of my PhD thesis based on these works can be found here and here.

At the moment, I am working towards incorporating affine models into our framework. A related longish-term goal is to use our construction together with the path integral formalism to quantise integrable field theories in a covariant manner.

Causal set theory

As a visiting scholar at the Raman Research Institute, Bengaluru, I worked on causal set theory, an approach to quantum gravity that postulates that spacetime is fundamentally discrete and replaces the spacetime continuum with locally finite posets. If one were to randomly pick a poset from the space of all finite posets, it would far more likely be non-manifold-like. This poses a challenge to causal set theory, since one expects continuum-like dynamics to arise in its classical limit. A result from my collaboration with Dr. Abhishek Mathur and Prof. Sumati Surya helped establish that certain classes of non-manifold-like causal sets are suppressed in the causal set path sum despite being more typical that manifold-like ones, making the emergence of manifold-like behavior possible without any restrictions.

Entropy and the link action in the causal set path-sum
Abhishek Mathur, Anup Anand Singh, and Sumati Surya
Classical and Quantum Gravity
arXiv:2009.07623 [gr-qc]
December 2020

Black holes, information scrambling, and quantum chaos

For my MS thesis work done under the supervision of Prof. Spenta Wadia at the International Centre for Theoretical Sciences (ICTS-TIFR), Bengaluru, I tried to better understand the scrambling of information by black holes using ideas from holography, quantum information and chaotic dynamics. I focused, in particular, on the Sachdev-Ye-Kitaev (SYK) model, a strongly coupled, quantum many-body system that is maximally chaotic, nearly conformally invariant, and exactly solvable, properties that make it interesting from the point of view of holography. A new result from my MS project was a derivation of the effective action of a charged version of the SYK model achieved by reducing the original theory of Majorana fermions to a theory of bilocal fields.

Chaos in field theory and gravity
Anup Anand Singh
Supervisor: Spenta Wadia
MS thesis
May 2018